Certainly! Let's solve the quadratic equation [tex]\( -x^2 - 8x - 16 = 0 \)[/tex] step by step.
### Step 1: Identify the coefficients
For the quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex], we identify the coefficients:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = -8 \)[/tex]
- [tex]\( c = -16 \)[/tex]
### Step 2: Calculate the discriminant
The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by [tex]\( \Delta = b^2 - 4ac \)[/tex].
[tex]\[
\Delta = (-8)^2 - 4(-1)(-16)
\][/tex]
[tex]\[
\Delta = 64 - 64
\][/tex]
[tex]\[
\Delta = 0
\][/tex]
### Step 3: Calculate the square root of the discriminant
Since the discriminant is zero, its square root is also zero.
[tex]\[
\sqrt{\Delta} = \sqrt{0} = 0
\][/tex]
### Step 4: Use the quadratic formula to find the roots
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substitute the values of [tex]\( b \)[/tex], [tex]\( \Delta \)[/tex], and [tex]\( a \)[/tex] into the formula:
[tex]\[
x = \frac{-(-8) \pm 0}{2(-1)}
\][/tex]
[tex]\[
x = \frac{8 \pm 0}{-2}
\][/tex]
Since [tex]\(\pm 0\)[/tex] does not change the value, we have:
[tex]\[
x = \frac{8}{-2}
\][/tex]
[tex]\[
x = -4
\][/tex]
### Step 5: State the solution
Since the discriminant is zero, there is only one unique solution (a repeated root).
[tex]\[
\underline{x} = -4
\][/tex]
Hence, the solution to the equation [tex]\( -x^2 - 8x - 16 = 0 \)[/tex] is [tex]\( x = -4 \)[/tex].
